Proof That the Plane With the Usual Topology is Second Countable
Theorem
The plane with the usual topology is second countable.
Proof
A topological space
is second countable if a countable basis exists for the usual topology.
The topology for
with the usual metric is the set of open balls with the Euclidean metric. Take
as the set of open balls with centres whose coordinates and radii are both rational. The set rational numbers is countable, as is the cartesian products of rational numbers (the cartesian product of countable sets is countable). Henceis countable.
Hence
defined in this way is second countable.
